Question

After traversing a wireless error filled channel packets arrive at a network node according a

Poisson Process with rate

λ

. These packets could have corrupted bits due to the wireless

channel which has a bit error probability

p

. A packet is said to arrive at the node successfully

if no bits are in error. The number of bits in a given packet is Poisson distributed with mean

μ

. Assume that packets are not retransmitted. Find the rate at which successful packets

arrive at the network node.

Answer 1

Let X be the number of bits in a given packet. Then, X ~ Poisson($\mu$)

So, probability of a success packet = Probability that no bits are in error in a packet =  (1 - p)^X

Let Y be the number of packets arrive at a network node. Then Y ~ Poisson($\lambda$)

Then, Number of success packets = probability of a success packet * Y =  (1 - p)^X Y

Expected number of success packets = E[(1 - p)^X Y]

= E[(1 - p)^X ]E[Y] (X and Y are independent events)

= $e^{-p \mu}$ * $\lambda$ (By the probability generating function of a Poisson random variable with rate parameter $\mu$, E[z^X] = $e^{\mu (z-1)}$ )

= $\lambda$$e^{-p \mu}$